x(t) t
x t
w(a) a
s
s(t) =
x(a)w(t − a)da
s(t) = (x ∗ w)(t)
w
w 0
x
w
t
x w t
s[t] = (x ∗ w)(t) =
∞
a=−∞
x[a]w[t − a]
I
K
s[i, j] = (I ∗ K)[i, j] =
m
n
I[m, n]K[i − m, j −n]
s[i, j] = (I ∗ K)[i, j] =
m
n
I[i − m, j − n]K[m, n]
m
n
s[i, j] = (I ∗ K)[i, j] =
m
n
I[i + m, j + n]K[m, n]
a b c d
e f g h
i j k l
w x
y z
aw + bx
+ ey + fz
bw + cx
+ fy + gz
cw + dx
+ gy + hz
ew + fx +
iy + jz
fw + gx +
jy + kz
gw + hx
+ ky + lz
Input
Kernel
Output
m n
m × n O(m × n)
k k × n
O(k × n)
k
m
x
1
x
2
x
3
s
2
s
1
s
3
x
4
s
4
x
5
s
5
x
1
x
2
x
3
s
2
s
1
s
3
x
4
s
4
x
5
s
5
X
3
3 X
3
X
3
O(k×n)
k k
m m n k
m × n
x
1
x
2
x
3
s
2
s
1
s
3
x
4
s
4
x
5
s
5
x
1
x
2
x
3
s
2
s
1
s
3
x
4
s
4
x
5
s
5
S
3
S
3
3 S
3
S
3
x
1
x
2
x
3
h
2
h
1
h
3
x
4
h
4
x
5
h
5
g
2
g
1
g
3
g
4
g
5
x
1
x
2
x
3
s
2
s
1
s
3
x
4
s
4
x
5
s
5
x
1
x
2
x
3
s
2
s
1
s
3
x
4
s
4
x
5
s
5
f(x) g f(g(x)) = g(f(x))
g
g
g(x) i g(x)[i] = x[i − 1] x
x
x
319 ×280 × 3 = 267, 960
320 ×280 × 319 × 280
2 × 319 × 280 = 178, 640
Convolutional Layer
Input to layer
Convolution stage:
Affine transform
Detector stage:
Nonlinearity
e.g., rectified linear
Pooling stage
Next layer
Input to layers
Convolution layer:
Affine transform
Detector layer:
Nonlinearity
e.g., rectified linear
Pooling layer
Next layer
Complex layer terminology Simple layer terminology
k
k
0.1 1. 0.2
1.1. 1.
0.1
0.2
... ...
... ...
0.3 0.1 1.
1.0.3 1.
0.2
1.
... ...
... ...
DETECTOR STAGE
POOLING STAGE POOLING STAGE
DETECTOR STAGE
0.1 1. 0.2
1. 0.2
0.1
0.1
0.0 0.1
K
i,j,k,l
i
j k l
V
i,j,k
i j k
Z
i,j,k
=
l,m,n
V
l,j+m,k+n
K
i,l,m,n
l m n
s
c
Z
i,j,k
= c( , , s)
i,j,k
=
l,m,n
[V
l,j×s+m,k×s+n
K
i,l,m,n
] .
s
m×m
k ×k m−k+ 1×m−k +1
1 × 1
k
m+k −1 ×m+k −1
... ...
...
...
...
...
...
...
...
i j k
l m
n
Z
i,j,k
=
l,m,n
[V
l,j+m,k+n
w
i,j,k,l,m,n
] .
i l
m
n m n
k
t t
Z
i,j,k
=
l,m,n
V
l,j+m,k+n
K
i,l,m,n,j%t,k%t
s
c( , , s)
J( , ) c
J
G
i,j,k
=
∂
∂Z
i,j,k
J( , ).
g( , , s)
i,j,k,l
=
∂
∂K
i,j,k,l
J( , ) =
m,n
G
i,m,n
V
j,m×s+k,n×s+l
.
h( , , s)
i,j,k
=
∂
∂V
i,j,k
J( , ) =
l,m|s×l+m=j
n,p|s×n+p=k
q
K
q,i,m,p
G
i,l,n
.
h
= h( , , s).
g( , , s)
c( , , s)
g c h
k × k
i,j,:
=
(i, j)
i,j,:
2 × 2
d d
d
k
w
O(w
d
)
O(w × d)
k
•
•
•
•
•
I(x, y)
x X y Y
w(x, y)
s(I) =
x∈X
y∈Y
w(x, y)I(x, y).
w(x, y)
w(x, y; α, β
x
, β
y
, f, φ, x
0
, y
0
, τ) = α exp
−β
x
x
2
− β
y
y
2
cos(fx
+ φ),
x
= (x − x
0
) cos(τ) + (y −y
0
) sin(τ)
y
= −(x − y
0
) sin(τ) + (y −y
0
) cos(τ).
α β
x
β
y
f φ x
0
y
0
τ
x
0
y
0
τ
x y x
y
x
0
y
0
τ
x
y
w
x
α exp
−β
x
x
2
− β
y
y
2
x
y
α β
x
β
y
x
0
y
0
τ
x
0
y
0
τ
x
0
y
0
τ
beta
x
β
y
β
x
β
y
β
× f
φ f φ
φ f ×
cos(fx
+φ)
x
f
φ
L
2
c(I) =
s
0
(I)
2
+ s
1
(I)
2
s
1
s
0
φ φ s
1
s
0
s
0
s
1
I(x, y) exp(−β
x
x
2
− β
y
y
2
)
f τ (x
0
, y
0
)
τ